In 1859, Georg Friedrich Bernhard Riemann had announced the following conjecture, called the Riemann hypothesis : The nontrivial roots (zeros) $s=sigma+it$ of the zeta function, defined by: $$zeta(s) = sum_{n=1}^{+infty}frac{1}{n^s},,mbox{for}quad Re(s)>1$$ have real part $sigma= 1/2$. In this paper, it was used the expression of $zeta(1-bar{s})$ in function of $zeta(s)$, it implies to the proof that $Re(s)=sigma= 1/2$.

This work falls within the scope of recreational mathematics and the experimental exploration of the order-3 magic square of squares. Instead of resolving the open problem of the existence of such an object, the research focuses on "near-miss" structures. The analysis is based on the reduction of the original problem to three three-term arithmetic progressions of square numbers with a common difference (r), whose first terms also form an arithmetic progression.By relaxing the condition requiring the initial terms to form an arithmetic progression, this paper introduces a percentage error function (p). This function allows for a quantitative assessment of the discrepancy between the differences of these terms, and consequently, a quantitative measure of how closely a configuration approximates a fully magic structure.Subsequently, an algorithm developed for this study was applied to generate all three-term progressions of integer squares with a common difference r. To optimize calculations and eliminate redundant operations, a filter based on the theory of quadratic residues in the ring Z_240 was implemented. Ultimately, a deterministic exploration of the differences r from 2 to 70,000,000,000,000 was conducted.Next, a geometric parametrization utilizing 120-degree Eisenstein triangles is presented. This enabled a dimensionality reduction of the error optimization problem to a function of a single real variable. It was demonstrated that the theoretical optimum point with zero error corresponds to an irrational number, preventing its direct realization in integers. It was proven, however, that despite structural limitations, it is possible to construct a parametric matrix generating semi-magic squares with equal sums in rows, columns, and on one diagonal. Diophantine approximation and continued fractions yielded an infinite sequence of rational approximations where the error function converges asymptotically to zero, while the ratio of the diagonal sums remains close to 1. This is illustrated by a numerical example where the ratio yields a value on the order of 1 - 7.3506 x 10^(-103230).The results offer a new method for the assessment of near-miss structures based on the error function p. They outline an approach based on Eisenstein triangles, for generating configurations where the error p tends to zero and the diagonal sum ratio approaches 1. This work does not prove the non-existence of the magic square of squares, but provides a universal research framework for approaching ideal structures.

This paper presents an experimental protocol for testing whether an intermediate state in multi-digit addition has causal influence through hidden activations, visible reasoning tokens, both, or neither. Each arithmetic trace is generated by an executable interpreter and records the digit position, carry, partial result, next transition and final answer. The proposed test replaces a valid intermediate state and checks whether the remaining trace follows the corresponding counterfactual computation. Hidden-only, visible-token-only and joint interventions are compared with shuffled-donor, unrelated-donor and same-norm random controls at three positions around state-token emission. Evaluation uses full-trace counterfactual transition fidelity, nuisance invariance, repair, donor leakage and collateral change on separately held-out IID and out-ofdistribution splits. Attribution methods may identify candidate locations, but every retained location must pass an exact activation intervention. This paper reports the protocol and no new model results. Behavioural accuracy and a readable rationale alone cannot establish causal authority; the proposed interventions test it on held-out data.

When a geometric modification of spacetime is equivalent, by a field redefinition, to a theory we already know, does anything physical survive the equivalence? We examine a metric-compatible contortion of the form $K^lambda_{ muu} = F(phi),epsilon^{lambdaho}_{ muu},partial_hophi$, generated by the gradient of a scalar field and totally antisymmetric in its three indices, which is unique within its class once one requires that spinless particles follow the geodesics of general relativity. The Ricci scalar acquires a correction $R(Gamma) = R({}) + 6F^2(partialphi)^2$ that shifts the scalar kinetic term by a factor $1 - 6F^2/kappa_g$, yielding a ghost bound $M_F > 0.49,M_{m Pl}$, and the Schwarzschild metric remains an exact vacuum solution when the scalar field is constant. We show that the bosonic sector is equivalent, by a field redefinition, to a canonical scalar field minimally coupled to general relativity, so that the modification of the Kalb-Ramond duality is a statement about the parametrization rather than about the physical spectrum. The survival of a residual coupling in the matter sector is not specific to this construction, since it also occurs in the Jordan-Einstein frame equivalence of scalar-tensor theories, but the form of the surviving coupling is. Here the residual coupling is parity-odd, spin-dependent, and non-universal, affecting only matter with spin. We estimate the magnitude of this force in a cosmological background and find it to be far below observational sensitivity; we do not claim that it becomes observable in regimes that we have not calculated.

This article presents examples of linear partial differential equations with analytic coefficients and system of such equations for which the solutions to the Cauchy problem in the class of continuously differentiable functions are non-unique. More precisely, the time during which the analytical solution to the Cauchy problem remains unique is finite for such equations and system,after which this solution may even transform into one of many possible solutions. These solutions may have unusual properties.Key words: systems of linear partial differential equations; analytical coefficients; Cauchy problem; Holmgren’s theorem; uniqueness of solution.

In this talk, we will introduce the importance of the division by zero containing the division by zero calculus and its great impact to elementary mathematics and mathematical sciences for some general mathematicians. For this purpose, we will give its global viewpoint in a self-contained manner by using the related references. (The Third Lecture at Peking University on 5 October, 2026).

This paper establishes a rigorous real-analytic and geometric framework governing the distribution of the non-trivial zeros of the Dirichlet eta function $eta(s)$ across the critical strip $mathcal{S}={sinmathbb{C}:0<mathbb{R}(s)<1}$. By decomposing $eta(s)$ into its strictly positive, monotonic real-axis baseline profile $eta(sigma)$ and two alternating quadratic half-angle residual series, $S_{Re}(sigma,t)$ and $S_{Im}(sigma,t)$, we demonstrate that the vanishing condition $eta(s_{0})=0$ at $s_{0}=sigma_{0}+it_{0}$ forces the dynamic residual series to collapse identically onto the static real baseline: $S_{Re}(sigma_{0},t_{0})=S_{Im}(sigma_{0},t_{0})equiveta(sigma_{0})$ Combined with functional reflection, any non-trivial zero mandates a simultaneous coincidence at the symmetric coordinate $1-sigma_{0}$ Utilizing the Mellin transform representation of the Fermi-Dirac integral kernel, we prove that the baseline disparity function $A(sigma)=eta(sigma)-eta(1-sigma)$ is strictly monotonically increasing on (0, 1) with its unique root located at $sigma=1/2$. Consequently, at $sigma=1/2$, the baseline disparity vanishes identically $(A(1/2)equiv0)$, naturally forming an invariant critical node where all residual profiles intersect. For any hypothetical zero off the critical line $(sigma_{0}e1/2)$, the system imposes a stringent dual level-crossing requirement: the dynamic residual oscillations must bridge a deterministic, strictly positive real gap $A(sigma_{0})>0$ simultaneously across dual orthogonal projections with an exact $pi/4$ phase shift. We establish that this simultaneous locking condition enforces an overdetermined phase alignment and a structural polarization disparity between the dual channels, providing an absolute real-variable obstruction that strictly confines all non-trivial zeros to the critical line $mathbb{R}(s)=1/2$. By demonstrating that this geometric barrier is mathematically insurmountable, this work establishes a rigorous and definitive proof of the Riemann Hypothesis.

The general theory of relativity fundamentally relies on the metric tensor to describe gravitational phenomena, traditionally eschewing the vector field formulations ubiquitous in electrodynamics and quantum field theories. This study introduces a novel framework that defines the spacetime metric and its inverse via a set of underlying vector fields. By expressing the metric tensor in terms of these fields, we derive the corresponding geodesic and Einstein field equations, revealing striking structural parallels with electrodynamics—most notably, the natural emergence of a covariant Lorentz-like force and a stress-energy tensor mathematically analogous to that of electromagnetic fields. To demonstrate the astrophysical viability of this approach, we solve the equations for a spherically symmetric, static mass distribution, yielding a non-singular alternative to the Schwarzschild metric. The practical validity of this metric is subsequently tested through the recalculation of the perihelion precession of planets and the gravitational deflection of light. The results provide precise agreement with perihelion precession observations while predicting a slightly modified value for light deflection (1.90 arcsec), offering a potential theoretical basis for the historical discrepancies observed between visible light and radio wave measurements during solar eclipses.

The Lorentz factor γ(β) =1/ √︁(1 — β^2), where β = v/c, is normally considered for real |β| < 1. Allowing its argument to range over the complex plane gives Γ(z) =1/√(1 — z^2), whose real and imaginary parts generate a striking system of level curves organized around the branch points z = +/-1. The Cauchy—Riemann equations explain why the regular curves from the two families intersect orthogonally. More surprisingly, eliminating the square root shows that each nonzero contour lies on a degree-eight algebraic locus. This apparent complexity is deceptive. Solving instead for z gives z = +/-√︁ (1 — Γ^-2), revealing that the apparently complicated contours in the z-plane are locally the preimages of vertical and horizontal straight lines in the Γ-plane. Thus, a familiar real function from special relativity provides an elementary example in which complicated algebraic contours conceal a simple conformal geometry.

A diffraction observation scheme is proposed that makes it possible to investigate the influence of the shape and size of the undisturbed part of a quasi-parallel light beam on the spatial position of the elements of the diffraction pattern. Experimental observations of the independence of the positions of the pattern elements from thisparameter were performed in the range from 0.01 to 2 mm.

We present Difference Power Space Extrapolation (DPSE), a geometric method for computing the n decimal digits of π by combining multiple polygon observations in difference power space. The paper shows that DPSE should be understood as a higher-order development of the Difference-Power Cumulative Method (DPCM), rather than asan unrelated construction. For the polygon semiperimeter surrogate Sk = nksin(π/nk), the geometric increments ∆Sk = Sk+1 − Sk admit a structured boundary-layer expansion that can be cancelled by Richardson-type linear combinations. For fixed extrapolation orderW, this yields the error law Ek,W = O(4−(W+1)k), while the diagonal Neville sequence exhibits an approximately quadratic error exponent in the physical observation index k. Consequently, the required physical observation count for n correct decimal digits is reduced from lineargrowth to order O(√n), while preserving the low-memory character of geometric iteration. Numerical experiments include a direct milliondigit fingerprint verification.

At first, I will introduce simply the analytical and classical reproducing kernels on the complex plane. Here, firstly, we recall the representations of reproducing kernels in terms of the theta functions that were obtained by D. A. Hejhal, J. D. Fay and A. Yamada. Indeed, the materials stated here are in Yamada's papers.Then, we will consider their meanings from the viewpoints of reproducing kernel theory, in particular, from the viewpoint of inequalities. We will find many important open problems. (The Second Lecture at Peking University on 3 October, 2026)

The Multiple Zeta Value zeta(2,1) is a double sum over two positive integer indices. If these indices are restricted by parity to cover either only even or only odd positive integers, the Multiple Zeta Value splits into four non-overlapping subseries. The manuscript evaluates these four series in terms of the Riemann Zeta function, pi and log 2.

This paper presents a consistent, mathematical-geometric framework describing quantum mechanics, gravitation, and cosmology within a "Bohmian" quantum field theory outside the standard model. By coupling a real helical sub-quantum domain (Helical Spin Theory, HST) with a space-filling background medium in a dynamic tachyonic momentum mode (Higgs-Tachyon-Push Field Theory, HTP), a quantitative reproduction of key astrophysical and quantum mechanical benchmark tests is achieved as an asymptotic limiting case within an absolute Euclidean reference frame. Gravitational light deflection, Mercury's perihelion precession, and the anomalous Pioneer Doppler drift are derived without invoking spacetime curvature via a spat-temporally Variable Speed of Light (VSL). On the microscopic scale, the double-slit paradox is formalized using an ontic, non-local pilot wave in line with De Broglie-Bohm mechanics. On the cosmological scale, the observed Hubble tension is mathematically resolved as a historical averaging effect derived from a degressive mass accumulation rate . Structurally, the HST is extended to spin-1/2 fermions; via a topological Möbius helix symmetry and a hyperbolic pilot-wave equation, the real negative mass squared of the electron neutrino measured by the KATRIN collaboration is rigorously derived as a geometric media-coupling parameter, providing a media-mechanical solution to neutrino oscillations. The model thus provides a closed mechanical framework that substitutes the requirements for Dark Matter, Dark Energy, and virtual particles within the macroscopic limit.Official preprint registered and priority-protected via CERN Open Science Repository Zenodo under DOI: 10.5281/zenodo.23088551.

The identity x^2 + (x+1)^2 + (x(x+1))^2 = (x^2+x+1)^2 is the motivation for study of the equation d^2 + e^2 + n^2 = x^2, where d and e are conjugate divisors of n, that is, n=de and $d<e$. Let N_{n} be the set of solutions to the given equation. Solutions with |N_{n}|>1 fall empirically into two families: a triangular family mathscr{T}, and a Fibonacci family mathscr{F}, with virtually no overlap. It is conjectured that the two families comprise all n with |N_{n}|>1.

Trees are dynamically complex branched structures in which trunks, branches and higher ordercrown elements possess characteristic motions and are mechanically coupled. In a doctoral thesis completed in 1983 and formally awarded in 1984, Scannell proposed a qualitative mechanical principle termed aeroelastic similarity. Measurements on Sitka spruce suggested approximate relationships between modal frequencies at successive structurallevels, together with direct evidence of coupled branch motion. The original interpretationproposed that overlapping frequency ranges could permit wind-induced mechanical energyto be redistributed through the crown and dissipated, thereby restricting large-amplituderesonant motion. This paper reassesses that hypothesis against four decades of subsequent tree biomechanics research. Later work explicitly tested the named aeroelastic-similarity theory and subsequently developed mass damping, multiple resonance damping and damping by branching.The evidence does not support a universal hierarchy of exact modal equivalence. It does,however, support a broader principle in which frequency-compatible, mechanically coupled treemotions can exchange energy and thereby provide access to distributed dissipative mechanisms.We formulate this surviving idea as dynamically coupled modal compatibility, not modalidentity. A reduced coupled-oscillator model separates conservative energy redistribution fromdissipation, introduces descriptive measures of spectral separation and mode hybridisation,and motivates a hierarchical modal-network interpretation. We then propose a falsifiableexperimental programme based on controlled detuning in numerical models, laboratorybranched structures and living trees. Aeroelastic Similarity is therefore retained not asan exact frequency-scaling law, but as a testable hypothesis concerning the dynamicalorganisation of branched structures.

The theory of reproducing kernels is very fundamental, beautiful and has many applications in analysis and numerical analysis as in the Pythagorean theorem. In this first lecture, I would like to intoroduce some general viewpoint for reproducing kernels and some essences of reproducing kernels. However, the theory is already much more developing widely and deeply on this century. (The First Lecture at Peking University on 1 October, 2026)

We consider electron as the the Brownian particle in the sea of the blackbody photons.The bremsstrahlung force caused by the acceleration of electron is involved in Langevin equation,Then we solve the Langevin equation with the bremsstrahlung force. The analogue of an electron - muon - isconsidered as the possible theoretical and experimental activity concerning the Langevin equation

In this article, I introduce a new integration method based on algebraic tools. This new integral enables an optimal solution with polynomial complexity P to be obtained for an NP-hard problem. The optimization problem involves minimizing a quadratic form with binary variables. The optimization algorithm is presented at the end of the report. The project concludes with numerical simulations to demonstrate this new concept of an integral, and the R programs are provided in the appendix so that the reader can test the new concept. Applications of this optimization problem span diverse fields such as economics, finance, machine learning and stastical physics (the Ising Model).

This paper presents new formulae for the harmonic numbers of order k, Hu2096(n), and for the partial sums of two Fourier series associated with them, denoted here by Cᶻu2096(n) and Sᶻu2096(n). I believe this new formula for Hu2096(n) is an improvement over the digamma function, ψ, because it is simpler and stems from Faulhaber's formula, which provides a closed form for the sum of powers of the first n positive integers. We demonstrate how to create an exact power series for the harmonic numbers, a new integral representation for ζ(2k+1), and a new generating function for ζ(2k+1), among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature.

This paper presents formulae for the sum of the terms of a harmonic progression of order k with integer parameters, HPu2096(n), and for the partial sums of its two associated Fourier series, Cᶻu2096(a, b, n) and Sᶻu2096(a, b, n). HPu2096(n) is built from the ground up, with a power series for 1/(aj + b)ᵏ that is summed over j using Faulhaber's formula. These new formulae are a generalization of the formulae created in a previous paper and were achieved using a slightly modified version of the reasoning employed before.

In this paper, two novel schemes for teleportation and quantum secure directcommunication are proposed by using a new method for local quantum measurementdiscrimination (LQMD) (Laser Phys. Lett. 22 (2025) 125211). In the two schemes, two space-like separate observers (Alice and Bob) share two and N encoding-decoding sets (EDSs) respectively, each EDS consisting of 30 eightqubit GHZ states and there is no classical channel between Alice and Bob. It is shown that, if Alice and Bob agreed in advances that one of them (e.g. Alice) should measure her qubits via two types of measurements before an appointedtime T, and Bob measures his qubits and uses probability comparison method at the time T, two schemes can be completed without help of classical channel. This means that the superluminal quantum communication can be realized by using the LQMD.

String theory is not a quantum theory. It’s a classical theory in which "discrete quantization" is supposed to emerge by analogy to the various discrete vibration frequencies of a taut wire in a piano or guitar. The problem is this is a circular argument: the wire is composed of particles. The whole thing is riddled with error. What they’re doing is not quantum field theory, but classical field theory fiddled to get quantization. But it doesn’t work. Then they block any other avenue using peer-review dogma. The mainstream theory is "standard" because it’s "peer-reviewed" (just groupthink dogma enforced), everything else is non-standard and thus dangerous, which must be suppressed. String theory is a classical continuum model with quantization applied to it. It does not derive quantum field theory from particles; it quantizes classical geometry and declares the result quantized. This paperreplaces that classical-continuum error with a fully mechanistic, non-speculative, empirically grounded quantum-gravity framework derived entirely from measured quantities. (This version 2.0 replaces some of the sociological analysis in version 1.0 with a patient refutation step-by-step of key string theory tactics. The original version 1.0 is much briefer at just 20 pages, and less technical, and goes more deeply into the political and financial motivations of string theory as a result of Parkinson's Law. Some readers may want to read both versions, or version 1.0 first. There are possibly multiple ways to get the facts across, so it's not clear which version will have the most appeal.)

Claims of "scalar", "longitudinal" or "Tesla" electromagnetic waves often combine threedistinct phenomena: longitudinal electric-field components in anisotropic media, superluminalphase or group velocities, and an additional propagating scalar degree of freedom. Weseparate these questions and derive the relevant results from first principles. First, source-free Maxwell electrodynamics with tensor permittivity and permeability is reduced to itsplane-wave eigenvalue problem. The Gauss constraint is shown to make the displacementfield, rather than the electric field, transverse to the wave vector; an explicit anisotropicexample consequently has a predominantly longitudinal electric field and a phase velocityexceeding the vacuum light speed. The Poynting theorem is then derived for a dispersiveanisotropic medium, leading to the Brillouin energy density and energy velocity and showingwhy a superluminal phase velocity does not imply superluminal energy or signal propagation.Second, a genuine scalar degree of freedom is introduced through a Lorentz- and gauge-invariant dilaton-like action. All Euler—Lagrange equations are derived explicitly. Gauge,translation and Lorentz symmetries are treated with Noether’s theorem, yielding charge,energy—momentum and angular-momentum/boost conservation laws. Linearisation about astatic electric background gives the full mixed scalar—electromagnetic dispersion relation, itsnearly massless distinguished limit, and a scalar-associated branch which is purely longitudinalelectrically and has zero magnetic perturbation for propagation parallel to the backgroundfield. The same action gives a finite-electrode voltage-dependent capacitance; this predictionis derived using the Green function of the scalar equation. Existing scalar—photon andresonator searches already constrain the coupling strongly, making large effects in the minimalunscreened theory implausible. We finally formulate direct static and dynamical experimentaltests of the remaining longitudinal mode.

Prime numbers can be represented in many ways. This work focuses on representing them using other primes: p, q, and r as: p×q±r. The aim of this research is to explore interesting properties of such prime representation, including the smallest product of p, q, and r that can represent a given prime number and 4 related hypotheses that were formulated when exploring this topic.

This work briefly reviews the main achievements of theoretical particle physics obtained using approaches based on symmetry and topology, and then highlights the weighty errors present in these theories. Particle theories are very abstract and developed within the framework of physical mathematics, which has no solid physical basis. However, real particles and real physical processes require finding a real basis on which fundamental physics should stand, and this is very important. The primary mathematical structure must be such that it is possible to apply mathematical physics to describe the physical picture of the world in real parameters that are actually present in particles and their interactions. It is precisely such a mathematical theory that is described in this work, the suitable geometry and topology are revealed, and references are provided to the corresponding physical theories developed by the author within the framework of mathematical physics, which already have a number of reliable experimental confirmations.

This study begins with the recognition that the Standard Model particle-antiparticle classification lacks a universal physical principle that determines the particle orientation of each charge-conjugate pair while grouping the proton, neutron, and electron into the same particle sector. Motivated by the weak-isospin doublet structure of weak fermion transitions, we introduce a new weak-isospin-based quantum number, N=2T_3L, where T_3L is the third component of left-handed weak isospin. States with N > 0 are assigned to the particle sector, states with N < 0 to the antiparticle sector, and states with N=0 to an N-neutral sector. In this classification, the proton and neutrino are particles, whereas the electron and neutron are antiparticles. The representative weak, hadronic, stellar, and early-Universe processes examined here consistently preserve the total additive N balance while redistributing N among different species. For fermion species defined by their left-handed weak-doublet orientation, the electroweak charge assignments give N=2Q−(B−L), where Q, B, and L denote electric charge, baryon number, and lepton number. Hence, for charge-conserving processes, exact additive N conservation requires ∆(B−L)=0. Accordingly, the same Universe that appears strongly particle-antiparticle asymmetric under the conventional classification is reinterpreted in the weak-isospin-based classification as a Universe in which positive-N and negative-N components cancel globally, yielding the global N-symmetry condition N_tot=0. This work further proposes a possible particle physics mechanism for the species-level particle-antiparticle and baryon asymmetry problems. A proposed N-conserving 1 → 3 decay process such as 'n̄ →p + e^− + ν_e' can generate, in a single decay event, the asymmetry direction required to account for the present cosmic particle abundance pattern, (δ∆p, δ∆n, δ∆e, δ∆ν) = (+1, +1, +1, +1). Related 2 → 2 scattering processes such as 'n̄ + e^+ → p +ν_e' and 'n̄ + v̄u2091 → p +e- ' produce the same asymmetry direction. Accordingly, this work recovers global particle-antiparticle symmetry through the N=2T_3L classification and proposes a possible solution to the species-level particle-antiparticle asymmetry through new N-conserving particle-conversion processes. The proposed mechanism can be tested directly through searches for N-conserving 1 → 3 decay and 2 → 2 scattering channels and their CP-conjugate rate asymmetries, while cosmic neutrino asymmetry, B−L violation, and the nature of neutrinos provide additional tests of the broader framework.

In standard quantum field theory, the Ward—Takahashi identity expresses the quantum version of Noether’s theorem and guarantees the decoupling of unphysical ghost states, preserving unitarity. In this paper we reformulate the Ward identity inside a mechanistic, vacuum—polarisation—driven QFT, where the fundamental ensemble weight is real (e.g. cos S rather than exp(iS)), and where gauge bosons arise as coherent waves of routing bias in a discrete spinor network associated with the real forms of SL(4, C). We show that the Ward identity emerges as a dynamical constraint: the vacuum couples only to conserved gauge currents, and the vacuum—polarisation energy functional is strictly gauge invariant. As a consequence, ghost—like, non—conserving excitations cannot couple to the vacuum and therefore have zero physical amplitude. This provides a physical, mechanistic implementation of the Ward identity compatible with discrete SL(4, R) spinor routing, finite subgroup quantisation (including Wilson’s H288), and the continuous SU(4) gauge structure arising in the coarse—grained limit.

The prevalence of sexual reproduction in nature remains an evolutionary paradox, given its substantial costs compared to asexual alternatives. While genetic explanations such as the Red Queen hypothesis emphasize coevolution with parasites, the role of spatiotemporal discreteness and population structure has been underexplored. Spatial structure matters because real populations are finite, interact locally, and are subject to demographic noise, factors that can fundamentally alter evolutionary outcomes. Here, we show that the discrete, nonlinear character of interactions between hosts and pathogens, when combined with spatial diffusion in low-dimensional habitats, naturally favors sexual reproduction. Using stochastic reaction—diffusion models and renormalization group analysis, we demonstrate that asexual populations face certain extinction in dimensions d≤2 due to discreteness-induced extinction (DIE), whereas sexual populations can persist and form stabilizing clusters. The mechanism hinges on the recurrence of random walks in low dimensions, which ensures pathogen-host encounters for asexuals but promotes protective clustering for sexuals. Our results suggest that the advantage of sex may emerge from purely physical and demographic constraints, independent of genetic mechanisms, thereby offering a new perspective on this classic evolutionary puzzle.

The time-dependent electric Aharonov-Bohm effect, where shielded potentials vary while electrons are in transit, offers a unique test of the Lorenz condition $partial_mu A^mu=0$. We introduce a phenomenological coupling $kappa$ between the scalar $partial_mu A^mu$ and the electron. In the field-free interior of shielded cylinders, $partial_mu A^mu = dot{Phi}/c^2$, giving an additional phase $Deltaphi_{lorenzScalar} = -kappa[Delta V(T)-Delta V(0)]$. For sinusoidal modulation this yields a $1-cos(omega T)$ term orthogonal to the standard $sin(omega T)$ Aharonov-Bohm phase. We propose a realistic electron interferometer, analyse fringe-field systematics, and project a sensitivity $|kappa|sim10^{-6},text{V}^{-1}$, the first such probe. A null result would confirm the scalar mode's decoupling from matter; a positive signal would uncover a new electromagnetic degree of freedom.

This paper proposes a new representation of integers, revealing significant information regarding their divisibility. Based on this novel integer representation, it provides necessary and sufficient conditions for numbers of the form $n^2+1$ to be prime. Meanwhile, the Goldbach's conjecture has been proven.

The Lorentz-FitzGerald contraction is the physical phenomenon where an object becomes contracted along the axis of motion. We will test this hypothesis by observing the movement of the Earth relative to the CMB rest frame.

We calculate the number of ways to arrange the ships in the game "Battleship." The answer: On a 10 × 10 grid, there are 26,509,655,816,984 (approximately 26.5 trillion) ways to place one ship of length 5, two ships of length 4, three ships of length 3, and four ships of length 2, with spaces between them. Rotations and reflections are not excluded from this count.

Taking all elements of physical reality to be beables (things that exist), our local-realistic study of Bell (1964) rests on two core principles: Einstein-locality (no beable travels superluminally) and Bohr-realism (some beables change interactively). This note (with expectation E, probabilities P, spin-half particles p), exposes the physical laws that correct Bell's error and refute his theorem.

An addition chain of length $h$ that leads to a number $n$ is a sequence of positive integers $s_0=1,s_1=2,ldots,s_h=n$ such that $s_i=s_j+s_k$~($i>jgeq k$) for each $1leq ileq h$. We introduce a matrix-theoretic framework for studying the properties of arbitrary addition chains that lead to a target integer $ngeq 2$ by associating each chain and the sequence of dominant and lower weight summands that calculate each term in the chain an adjacency matrix that encodes the internal geometry. In this linear-algebraic and matrix theoretic framework, we investigate how rank, rank profiles, singular values, eigenvalues, matrix norms, predecessor depths, and track-interaction terms encode the combinatorial and arithmetic structure of addition chains.

Black Hole Cosmology has traditionally been challenged by the apparent incompatibility between black hole interiors and cosmic expansion. However, because classical general relativity, on which this criticism is based, is itself incomplete in black hole interiors, the criticism rests on an incomplete premise. We reformulate this problem using total gravitational self-energy (GSE). The total GSE functional U_{gs−T}(R) = − (βGM_fr^2/R)[1 − (5β/7 )GM_fr/ Rc^2] contains a negative GSE term and a positive matter-GSE interaction term responsible for the repulsive radial response, where β denotes the structural coefficient of the GSE functional. For a matter-filled black hole interior, this energy landscape yields finite radii, including R_gs=(5β/7)R_S, R_{min,BH} = 5βR_S/(14−5β), and R_{min,E}=(5β/14)R_S. Thus, the point singularity is replaced by a finite-radius structure, allowing sufficiently massive black holes to contain macroscopic nonsingular interiors. This supports the possibility of closed black hole universes whose interiors exhibit weak local curvature and small tidal gradients. For our observable universe, the relevant configuration is not an isolated vacuum black hole, but a cosmological black hole embedded in a much larger, nearly homogeneous matter distribution. The matter-sector mass enclosed within the present particle horizon gives a Schwarzschild-equivalent event horizon radius R_S≃150.4 Gly, whereas the causal radius is R_U≃46.5Gly, giving R_S/R_U≃3.23 > 1. In this compactness sense, the observable universe lies within its own Schwarzschild-equivalent radius. The cosmological application of total GSE yields the dark energy density ρ_{Λ_m}= (βρ_m/2)(R_S/R_U)[(5β/14)R_S/R_U − 1]. Under the matter-dominated background approximation with w_{Λ_m}≃−1, the condition for accelerated expansion is ρ_{Λ_m} > ρ_m/2, or equivalently, R_U< R_{crit,U}≃0.279 βR_S. The framework provides a three-stage interpretation of cosmic history: early accelerated expansion driven by total GSE, a decelerating phase during the radiation era and the matter era, and late-time accelerated expansion driven by total GSE dark energy. Thus, accelerated cosmic expansion is not evidence against Black Hole Cosmology, but rather a central signature of a cosmological black hole. The predicted dark energy density provides a direct observational test of this interpretation of the observable universe.

This work introduces a discrete geometric model for the construction of a circle, basedon the hypothesis of a quantized space and the strict exclusion of the analytical continuum. The circular domain does not emerge as the limit of regular polygons, but rather from the progressive development of elementary circular sectors of increasing angular width associated with a step-by-step elongation of discrete radius; these sectors extend from the vertical axis of the circle, structuring their evolution into a numerical staircase pattern. A rectification procedure for the profile of thisstaircase is defined by means of a cutting line passing through the midpoints of the individual steps, thereby isolating right-trapezoid-shaped sections equivalent to the circular sectors of the discrete circle. The ordered accumulation of these elements reduces the measurement of the enclosed area to that of a corresponding equivalent isosceles right triangle. Under specific conditions of structural proportionality—and using a geometric equivalence constant (γ = 5/2) derived from the intrinsic properties of the configuration—it is demonstrated that the construction of this specific discrete model is governed by the exact dimensionless constant k = 25/8, which operates independently of the traditional analytical value of π.

Chang introduced Carlitz multiple polylogarithms (CMPs), generalizing in positive characteristic the Carlitz polylogarithms of Anderson and Thakur, and established a linear independence result for their ∞-adic algebraic values, from which follows the function field analogue of Goncharov's direct sum conjecture for multiple zeta values u200bu200b(MZV). Chen showed the analogue of this result in the v-adic setting, under the assumption that the place v is of degree one. The aim of this article is to show that Chen's result is valid for any finite place.

The usual power series sumalpha_{n}x^{n} are generalized by the substitution xto g(x) to power series built from a function sum a_{n}g^{n}(x). The key feature of our approach consists in using functions g with free parameters g(x)=g({p_{i}},x); the resulting expansions thus keep the Taylor—like behavior (i.e., they match derivatives at the expansion point), but can also be tuned to adjust the far—away behavior. We present six specific cases g_{X}, Xin{A,dots,F}, each containing five free parameters. We use examples to demonstrate that the generalization can improve the convergence (rate and domain), mimic branch points and branch cuts and have other interesting properties. Because the partial Bell polynomials are a necessary ingredient in our method, our results can be interpreted as new formulas for specific values of the Bell polynomials.

In this paper, the submicroscopic deterministic concept developed by the authoris applied to the problem of the neutrino mass. A particle appears from spaceconsidered as a mathematical lattice of primary topological balls, and induces adeformation coat in its surrounding. The principles of the interaction of particleswith space and through space between themselves are considered in detail. Theapproach states that real quarks possess only an integer charge (±e) and whenmoving they periodically change to the monopole state (⇄g) and hence, canonicalparticles are dynamic dyons. A neutrino emerges as a squeezed quark when it isin a monopole state, or in other words, the quark monopole state (a bubble in thetessellattice) is transferred to the appropriate lepton monopole state (a speck in thetessellattice). The self-mass (a ‘rest’ mass) for each neutrino flavour is calculated.The calculated value of the self-mass for the electron anti-neutrino is 1.222874 × 10^{−36} kg = 0.69893 eV/c^2. The concept of neutrino oscillations is revised, and another postulation is proposed, namely, that the transition from lighter to heavier flavours is due to the inelastic scattering of neutrinos on oncoming scatterers. As a result, the neutrino captures the mass defect, becomes heavier, and therefore the transitions ν_e → ν_µ and ν_µ → ν_τ occur; thus, the number of light neutrinos decreases in the neutrino flux studied.

Real physical space, as a purely mathematical structure built according tothe rules of set theory, topology, and fractal geometry, was proposed by MichelBounias (1943-2003) and the author. It emerges as a mathematical lattice ofprimary topological balls, which was named a tessellattice, and the size of acell/ball in the tessellattice is comparative with the Planck length, ∼ 10^{−35} m.Discrete fractal properties of the lattice allow the prediction of scales at whichsubmicroscopic to cosmic structures should occur. This approach allows thedevelopment of a submicroscopic concept of physics, which describes Natureat a much deeper level than offered by the quantum-mechanical formalismdeveloped at the atom scale, ∼ 10^{−10} m. In addition, the approach makesit possible to define such fundamental physical notions as mass and chargefrom first submicroscopic principles, and this actually means that fundamentalmathematics lays down the basic concepts of physics.

This paper examines moral responsibility for mistranslations of technical terms by large language models (LLMs) in light of expert contributions to public lexical resources. It argues that providing normatively appropriate translations in Wikipedia and Wiktionary may improve translation by LLMs that use these resources, and proposes the notion of deferentialism on terminological translation. It also questions the exclusive attribution of moral responsibility for mistranslation to model developers.

This article presents a proposed unified description of physical interactions within the framework of the special theory of relativity. The starting point is the principle of minimum potential energy, interpreted equivalently as the principle of minimum rest mass. According to this approach, every interaction can be described by a general force formula containing the charges of the given interaction, an interaction constant, and a dimensionless function of distance. The commonly used inverse-square dependence is treated here as an experimentally established approximation valid only within a limited range of distances, rather than as a universal law extending from zero to infinity.After the finite propagation speed of interactions is taken into account, interactions become one-sided: the force exerted by a source charge on another charge is determined by the state of the source at the point from which its interaction ray reaches the affected object. Therefore, the interaction must first be described in the rest frame of the source charge and then transformed to other inertial frames by means of Lorentz transformations. Using general vectorial Lorentz transformation formulas for U-spaces of arbitrary dimension, together with a relativistic form of Newton's second law of motion, unified vector formulas are derived for the acceleration and force acting on a point object carrying an arbitrary interaction charge. The appendix presents the derivation of the rest-mass derivative used in these formulas and provides numerical checks confirming the consistency of the transformations.

A constructive refinement of prime-type classification via nested primorial sievematrices. We introduce a primorial-based additive representation of integers (Mo numbers)and prove a residue separation property: for any prime q, primes in the arithmetic progressionq mod ps# cannot lie outside the nested Mo-type MTq. The Mo-types form a strictly descendinghierarchy MT2 ⊃ MT3 ⊃ MT5 ⊃ MT11 ⊃ ···, and we prove each deepest-type class containsinfinitely many primes (a constructive refinement of Dirichlet’s theorem without analytic densityestimates).We further classify twin prime candidates into coupled types such as MT(5,7) and MT(11,13),showing that the Twin Prime Conjecture is equivalent to the divergence of true twin pairs withinMT(5,7). As a rigorous corollary, Zhang’s bounded-gap theorem admits a Mo-type formulation:some even gap N ≤ 246 yields infinitely many prime pairs in a coupled Mo-type progression.The framework reframes classical distribution problems as explicit sieve-matrix counting,leaving open the quantitative density of twin pairs for future work

This essay traces the history of impedance matching as an overlooked core organizing principle in quantum physics. Impedance matching governs amplitude and phase of the flow of energy, of information transmission, in both classical and quantum mechanics. Although quantum mechanics retained the associated phenomena of amplitude, phase, reflection, transmission, resonance, tunneling, and scattering, it did not assimilate the organizing principle of impedance matching. Convergence with particle physics phenomenology and geometric algebra illustrates how ideas arising in separate traditions can unexpectedly meet. The essay reveals how a physical organizing principle can remain present in the phenomena while becoming largely invisible in the language used to describe them.

Many people dream of establishing a settlement on the planet Mars, especially the American entrepreneur Elon Musk. However, I believe this is nearly impossible because of the planet's extremely harsh environmental conditions. Compared with Mars, Earth is far more hospitable to human life.

In this paper we formally construct (mathbb{M}), a number system that contains three dimensions and forms a bridge between the complex numbers and the quaternions. In this system we define the operation of division by zero (1/0=infty). The third dimension of the system is the scalar infinity (s), which does not mix with real or complex numbers.Historically, a system with exactly three dimensions was impossible because of the multiplication of two new axes. In the quaternions these are (i) and (j), whose product must give (k), creating a new dimension because (ij) cannot be displayed in those three dimensions. Historically, taking the square root of (-1) was not allowed. This was solved by the new number (i), which introduced the complex numbers and achieved the transition from (mathbb{R}) to (mathbb{C}) by adding the new number (i) such that (i^2 = -1). (mathbb{R}) is one-dimensional, and (mathbb{C}) is a two-dimensional number system, so the question arises whether there are number systems above (mathbb{C}).The answer was first found by William Rowan Hamilton when he tried to construct a three-dimensional number system. By adding one new axis (j) such that (j^2 = i^2 = -1), he encountered the problem of what the product of (i) and (j) would give. He realized that it would give (k), a new spatial axis, and so he constructed a four-dimensional number system with axes (1,i,j,k) such that (k^2 = j^2 = i^2 = -1). Thus the three-dimensional algebra was skipped because no matter which axis we add, the product of two axes must always give something new, namely a fourth dimension.Later Frobenius and Hurwitz formalized this and proved that number systems are possible only in dimensions (2^n), where (n) is a natural number: namely 1 ((mathbb{R})), 2 ((mathbb{C})), 4 ((mathbb{H})), 8 ((mathbb{O})), and so on, up to 256 dimensions and beyond. This is the Cayley-Dickson construction, and the number systems obtained after the sedenions are no longer particularly interesting or useful for study because they lose critical properties such as associativity, alternativity, and commutativity.All these systems do not solve the problems for which new systems are introduced, namely the definition of undefined numbers. Just as the complex numbers define the square root of (-1), which was undefined in the real numbers, so the definition of (n/0) brings with it the definitions of other undefined numbers, such as the zeroth root and zero to the zeroth power.

Finite-time transport in the circular restricted three-body problem (CR3BP) can change qualitatively under perturbations to initial conditions, making robustness difficult to characterize using smooth trajectory sensitivities alone. This work investigates whether geometric distance to an outcome-change boundary can provide a usefulmeasure of transport robustness for a two dimensional family of invariant-manifold seeds generated from planar L1 Lyapunov orbits in the Earth—Moon CR3BP. A 160 × 33 parameter atlas comprising 5280 initial conditions was classified by first-passage outcome, and adaptive refinement produced 484 numerical samples of the resultingoutcome boundaries. To avoid arbitrary Euclidean distance in the family parameters, the seed-state map was used to induce a local pullback metric, defining a family-constrained boundary-distance diagnostic ρη. The metric construction was numerically consistent, and ρη revealed strongly nonuniform proximity to the resolved transportinterfaces. Its ordering, however, depended on the prescribed state weighting, with Spearman correlations ranging from approximately 0.794 to 0.999 across the tested metrics. Direct perturbation tests also rejected a guaranteed-radius interpretation: one outcome change occurred at approximately 0.5ρη, while finite directional searches at five representative points recovered no independent minimum flip distance. These results support ρη as a family- constrained geometric diagnostic of proximity to resolved finite-time outcome boundaries, but not as a certifiedoutcome-preserving radius or validated transport condition number. A stronger robustness measure requires converged boundary resolution, physically specified uncertainty geometry, and direct computation of minimum outcome-changing perturbations.

This paper explores the transition boundary where classical escape velocity thresholds intersectwith relativistic coordinate transformations. Using observational mass configurations from Sagittarius A*, we analyze how super luminal escape velocities (ve > c) force a geometric inversion ofthe local Minkowski metric tensor. We demonstrate that the information loss paradox is fundamentally a consequence of the future light cone compressing and tilting past the vertical axis, renderingoutward spatial paths geometrically non-existent to an external observer.

We report the detection of continuous microhertz waves originating from Jupiter, recovered from two decades of atmospheric pressure measurements across Germany. Barometers act as largeu2011aperture antennas for lowu2011frequency signals, allowing coherent recovery of a carrier near 55.97 $mu$Hz, which corresponds to twice the rotation frequency of Jupiter’s solid core. A directu2011conversion receiver iteratively compensates all known phase modulations caused by Earth’s orbit and the four Galilean moons, concentrating the wave’s energy into a narrow spectral line with a signalu2011tou2011noise ratio of 100. We measure the longu2011term frequency drift of Jupiter’s core and determine the modulation indices associated with each moon. The phaseu2011modulation pattern is stable across 318 independent realizations. Additional spectral features indicate that Jupiter hosts multiple internal oscillators with distinct frequencies. These results demonstrate that Earth’s atmosphere responds coherently to microhertz waves generated by nearby planets.

This paper reviews major approaches to the description of subatomic particles, suchas leptons, quarks, hadrons and nucleons. Among these approaches are quantum chromodynamics, soliton models, bag models and others. The main accent is on a theory of thereal physical space that acts as a scene on which all high-energy events take place. Wediscuss how a lepton and quark appear in the space constituted as a tessellation lattice ofprimary topological balls — the only structure that mathematics (i.e. set theory, topologyand fractal geometry) offers to the constitution of ordinary physical space. Since leptonsand quarks emerge in the tessellattice from a topological ball, they must interact withthis substrate. The principles of the interaction of subatomic particles with space andthrough space between themselves are considered in detail. The approach: i) states thatreal quarks possess the integer charge ±e and they periodically change to the monopolestate (hence, canonical particles are dynamic dyons); ii) naturally solves the problem ofconfinement of quarks; iii) reveals the dynamics of quarks in hadrons; iv) discloses an innerstructure of the proton and neutron; v) calculates the radius of the proton; and vi) derivesthe nuclear forces as the result of both direct coalescence of surfaces of the nucleons andthe overlapping of spatial excitations (named inertons) generated by the nucleons at theirmotion through the tessellattice. Experimental results showing nuclear transformationsin samples affected by artificially generated inerton fields are demonstrated.

This is an advanced introduction to the various types of normal random variables, with most of the needed preliminaries included. We first discuss the probability basics, standard central limits, and the theory of the usual, real normal variables, with examples, illustrations and numerous formulae. Then we go on a similar discussion regarding the complex normal variables, and the Rayleigh variables too. We then move to arbitrary dimensions, with a discussion regarding the Gaussian vectors, and related probability laws, featuring some functional analysis, and geometry and physics too. Finally, we provide an introduction to the quantum versions of the normal variables, and notably to those coming from free probability and random matrices.

Gödel's intuition accurately zeroed in on a core truth: that the expressiveness of non-trivial mathematical theories outpaces the efficacy of any inference engine. His paradoxical form of argumentation, however, begat immense confusion and pessimism, which this essay examines critically and aims to undo. His actual technical analysis exposed a historic oversight in classical set theory: the lack of internal structural variation within the Cantor sets $aleph_{0}$, $aleph_{1}$, etc. The paradoxical argumentation amounts to an indirect proof (reductio ad absurdum) of internal differentiation within $aleph_{0}$.This paper introduces a new concept of <i>structural grade separation</i> (SGS) within the Cantor sets and examines their foundational role and implications within formal mathematical frameworks, challenging the traditional conception of coarse set cardinality. By introducing a comprehensive SGS scheme, we demonstrate how structural stratification illuminates transfinite set theory without sacrificing logical consistency. The theoretical bounds of this construction against established axioms of arithmetic and set theory are analyzed, illustrating how explicit structural hierarchy resolves ambiguities inherent in standard set theory and cardinal assignments. The broader mathematical significance of graded set structures for decision algorithms and formal reasoning engines are explored.

The Pantheon + SHOES data is interpreted using transformed coordinates with distances that have origin at the beginning of the Universe. Distances (R) taken from the beginning of the Universe (R~0 and t~0) increase toward Earth and reveal a Newtonian Gravitational redshift (z<0.1). Distances determined in this way are the inverse of the luminosity distances measured from Earth. The revised coordinate frame yields a 1/R dependence of the redshift consistent with a Newtonian gravitational redshift. An alternative to the Hubble-Lemaitre expanding Universe is presented. The alternative Universe suggested is determined by Gravity, is static in size and contains Galaxies that show a Gaussian distribution of velocities. This is the model of the Universe first considered by Einstein.

According to a recently proposed vibrational interpretation of the Schrödinger equation for hydrogen-like atoms, stationary single-electron wave functions describe vibrations that are driven by phase waves associated with a bound electron moving on precessing orbits similar to orbits of the old quantum theory by Bohr and Sommerfeld. Based on this interpretation, the objective of this work is to provide insights into the structure of these wave functions by constructing similar wave functions from the corresponding electronic orbits and their associated phase waves without explicitly solving the three-dimensional Schrödinger equation. The proposed construction extends solutions of the two-dimensional Schrödinger equation into three dimensions and takes precession of the orbital angular momentum into account by a weighted average over all rotations about the quantization axis. The constructed wave functions are structurally similar to the standard wave functions of hydrogen-like atoms in spherical coordinates; thus, the proposed construction provides a new way of understanding the structure of these wave functions.

It is shown that all spin possibilities can be constructed by combining the fundamental spin 1/2 system a sufficient number of times. In order to express all spin possibilities mathematically we introduce the multispinor source. All component spins are on the same footing and additional symmetry requirements can be imposed on the multispinor. The multispinor description provides a unified approach to all spin values, with the first-order differential equation of spin 1/2 setting the pattern of field equations.

Covariance implies invariance under coordinate transformations, and its application within inertial reference frames enables the resolution of a wide range of relativistic spacetime problems. However, its application to gravitation restricts its use to spherical coordinate systems. Conventional transformation to a Cartesian system leads to issues regarding symmetry and commutativity, resulting in spacetime discontinuity due to the emergence of torsion.Coordinates lack physical significance, and the transformation between Cartesian and spherical representations should be trivial. Resolving the metric tensor in Cartesian coordinates via an imaginary group allows for the resolution of the Levi-Civita connection's torsion and, consequently, leads to a series of reconsiderations regarding spacetime.The structure of spacetime is revealed as a topological construction through Minkowski's hyperbolic foundation -- a formalism that allows for the representation of both the wave-like properties of gravitation and the metric characterization of electrodynamics.The normalized formalism proves productive both in demonstrating its consistency -- through the normalized formulation of the hydrostatic equilibrium of compact bodies, which supports the absence of singularities (as an alternative to the Tolman-Oppenheimer-Volkoff equation) -- and in its treatment of geodesics and rotating bodies (as an alternative to the Kerr metric). Formulation of the self-organization of spacetime via the normalized energy-momentum tensor, as well as cosmological issues regarding inertia, time and its irreversibility, entropic characterization, and large-scale behavior. Normalized metric is a continuous group that allows for the metric derivation of the Dirac equation, Yukawa coupling, and Hubble's law, while also offering insights into various conceptual issues, such as symmetry and vacuum polarization.

Modern natural-language systems usually represent linguistic meaning through statisticalmodels trained on large text corpora with substantial graphics-processor (GPU) resources.The present work examines a different, instrument-based hypothesis: that semantic differences between written assertions may be registered directly as differential torsion-fieldresponses, without training a statistical model. The study is formulated within Shkatov’storsimetry framework, in which the torsion field (TF) of physical objects and processes isrecorded as the torsional contrast (TC) between the measured object and the instrumentby means of single- and multi-coordinate torsimeters. To reduce spurious informationalphantoms that can accumulate in object—instrument—operator systems and affect repeatedmeasurements, we use the Method of Differential Test Assertions (MDTA). In this method,the object under test is composed from paired textual assertions with opposite meanings,and the differential torsion response between the two assertions is analysed. In 19 independent measurement sessions, a high-threshold assertion ("the total score is greater than θ"),treated as false under the experimental conditions, produced a higher torsional contrast thanthe paired true assertion ("the total score is less than θ") in every session. The mean paireddifference was +112.1 arbitrary units, and the exact one-sided sign-test probability for 19positive differences out of 19 was p = 2−19 ≈ 1.9 × 10−6. These results show a reproducibledifferential response between the paired textual assertions in the present protocol. Withinthe torsimetry framework, they support the working hypothesis that semantic content cancontribute to the measured torsional contrast of a written text; independent replication andexpanded controls will be required to determine the generality and physical mechanism ofthe effect.

Network of oscillators are studied to understand if graph (vertex) colouring can be obtained upon synchronization when a strong negative coupling is applied between oscillators when there is an edge present in the network and a small positive coupling is applied between oscillators when there is no edge. We find that oscillators with simple equations such as Thomas' systems are best suited for such couplings. We further observe that introducing a degree of variance in coupling strength improves synchronization to a great extent in favour of returning optimal graph colouring in such networks. We further study the role of parameters (which are obtained as a result of varying coupling strength) on synchronization. The present article thus ends up proposing a polynomial time heuristic method to solve the graph colouring problem.

A formula for the Hurwitz zeta function at the positive integers k, ζ(k, b), is createdby solving the real and the imaginary parts separately and then combining them. Afew different formulae for the Hurwitz zeta function are known from the literature, butthey are very general and usually hold for ℜ(k) > 1. The advantage of formulae thatonly hold at the positive integers is the fact that they are simpler and easier to workwith. P An analytic continuation of the generating function of ζ(k, b) is also obtained as k≥2 xk(ζ(k, b) − 1/bk), where the term 1/bk was subtracted for convenience.

A method to find prime factors of a given arbitrary number $n$ is proposed. When $n$ is semiprime, the method is shown to run in $O(n^2log(log(n)))$ time. Eigenvalue distribution of matrices utilized in the proposed method is conjectured.

We present new static solutions of SU(2) Yang-Mills equations for $n$ point sources aligned in a common isospin direction. The solutions possess a gauge field structure analogous to the BPST instanton while constituting a distinct class of static solutions, and satisfy the Coulomb gauge condition. They are either self dual or anti-self dual. Substitution of the solutions into the Yang-Mills equations yields localized source terms corresponding to a non-Abelian electric charge but also a magnetic monopole charge associated with each source. Furthermore, the resulting gauge fields generate a confining potential for quarks. These properties suggest that the present solutions provide a new class of nonperturbative configurations in Yang-Mills theory.

A Dirac equation for the $qbar{q}$ system is formulated in the presence of the confining potential derived from a static solution of the SU(2) Yang-Mills equations. The resulting radial equations define an eigenvalue problem with singular behavior at $r=0$ and $r=ho$, leading to a discrete spectrum of confined states.The system reduces to four sets of coupled radial equations for each of quark and antiquark components, which do not appear to admit closed-form solutions in terms of standard special functions. An approximate treatment is therefore developed to determine the corresponding energy spectrum. The calculated bound-state energies are found to be largely insensitive to the current quark masses and are of the same order as the observed meson masses, suggesting that the dominant contribution to the meson masses originates from the confining dynamics described by the present Yang-Mills solution. The quark and antiquark spectra exhibit an asymmetry associated with the gauge orientation of the underlying monopole-like configuration, although the individual energy eigenvalues are not gauge invariant observables.

We report a universal geometric identity for charged particles: the product of the classical electron radius r_e and the Schwarzschild radius r_s is fixed by the Planck length and the fine-structure constant: r_e r_s = 2 alpha l_P^2. The mass m completely cancels, indicating that this constraint is independent of the specific particle mass and represents a universal relation between charged matter and spacetime geometry. Two independent paths provide cross-validation: (1) an algebraic path—direct multiplication of the classical definitions; and (2) a dynamical path—Dekker (2014) derived the characteristic radius r_c = l_P sqrt(alpha) from exact solutions of the Einstein—Maxwell equations, which is algebraically equivalent to the identity above. From this identity, a natural length scale r_res = l_P / sqrt(2 alpha) is defined, corresponding to the Compton energy E_res = sqrt(2 alpha) E_P approx 1.47 x 10^18 GeV, which contains no free parameters and lies within a potentially testable window of ultra-high-energy cosmic-ray physics. We also propose an S^3 x R^3 fiber-bundle geometry as a heuristic framework providing a geometric picture for the resonance scale.

This essay explores a historical question: did twentieth-century physics gain extraordinary predictive power at the cost of losing a unified geometric language? It traces the development of Grassmann's exterior algebra and Clifford's geometric algebra, their eclipse during the rise of vector analysis, and the subsequent evolution of relativity, quantum mechanics, and gauge theory. This change in mathematical language also changed the direction in which theoretical physics sought explanation—from beginning with geometry and asking how physical law emerges, to beginning with physical law and deriving wavefunctions as solutions of field equations. It concludes by suggesting that recovering the geometric representation of the vacuum wavefunction may reopen a complementary bottom-up tradition in theoretical physics.

Much has been written about the notion of probability, and how this concept may be interpreted. In contrast, philosophers of science seem to have neglected the concept of (stochastic) distribution, although distributions are at least as fundamental. One the one hand, they generalize the basic mathematical notion of number, on the other, from a structuralist perspective, distributions precede probabilities and their interpretation.

Prototype based one-class learners such as Supervised Vector Quantization for One-Class Classification (SVQ-OCC) utilise squared Euclidean distance without metric adaptation and cannot adapt to the spatial structure of the data. Moreover, these learners are unable to capture relevant variation when the target distribution presents anisotropic structure in the feature space. This paper proposes One-Class Generalized Learning Vector Quantization (OC-GLVQ), whose discriminant substitutes the negative-class distance in GLVQ with a per-prototype visibility threshold, yielding a cost function that is jointly differentiable in prototypes, thresholds and metric parameters. A hierarchy of distance formulations from diagonal relevance weighting through global and local matrix projections to tangent subspace metrics is developed within the OC-GLVQ framework. The resulting classifiers adapt their geometry to the target distribution while retaining prototype-level interpretability. This paper identifies a gradient-magnitude mismatch between the prototypes and threshold parameters in high-dimensional settings and resolves it through a square-root reparameterisation. Experimental assessment on four practical datasets indicates the metric-adaptive variants consistently surpass both SVQ-OCC and Support Vector Data Description (SVDD), with local matrix adaptation yielding the strongest overall performance.

Goodman's new riddle of induction has mainly been treated in verbal terms. Its core concept, projectabiliy, has much in common with statistical estimation. Therefore, understanding when and why the latter approach works, also sheds much light on Goodman's riddle.

We propose that dark matter is not an exotic new substance, but a geometric necessity of four-dimensional spacetime. Prior to the electroweak Higgs phase transition, all fundamental particles were massless and propagated at the speedof light c. In this pre-Higgs epoch, the four dimensions of spacetime were fully equivalent— no dimension was geometrically privileged as "time". Upon Higgs condensation, the acquisition of mass froze each particle’s instantaneous propagationaxis as its individual time dimension, while the remaining three orthogonal dimensions became its spatial dimensions. Since exactly four basis dimensions are available, this produces four distinct dimensional sectors: one visible sector (our own) and three geometrically equivalent dark sectors. Particles across sectors share thesame four-dimensional manifold and thus interact gravitationally. Electromagnetic interactions— being confined to a shared three-dimensional spatial hypersurface— are geometrically forbidden between sectors. Dark matter thus requires no newparticles, no new forces, and no modifications to gravity: it is a direct consequence of the combinatorial geometry of (3+1)-dimensional spacetime. The predicted dark to-visible matter ratio of 3 : 1 is consistent with cosmological observations. The structural co-distribution of dark and visible matter emerges naturally from their shared gravitational manifold. Furthermore, the hypothesis provides a geometric explanation for the Hubble tension, predicts gravitational wave events without electromagnetic counterparts, and accounts for the persistent null results of all direct dark matter detection experiments.

John Bell’s 1964 article On The Einstein Podolosky Rosen Paradox concludes that a discrepancy between quantum theoretical and Kolmogorov-type expectation values is due to non-locality. On the other hand, this document concludes that the discrepancy is due to information loss rather than non-locality.

We consider an effective gravitational potential whose extra term is a regularised two-dimensional logarithmic potential — the logarithmic Green''s function of the two-dimensional Poisson equation. We derive its source surface density, Σ(R) ∝ rc^2/[(R^2 + rc^2)^2] (a two-dimensional Plummer-like distribution), and show that its total mass satisfies M = Vflat^2/G2 exactly, so that the model has a well-defined physical normalisation. The circular-velocity contribution is V2D^2(R) = Vflat^2 R^2/(R^2 + rc^2), which differs from a pseudo-isothermal (PI) halo in the approach to the flat asymptote (1/R^2 vs. 1/R corrections). We fit this model, MOND (two interpolating functions), NFW, PI and Burkert halos to the SPARC sample of 175 disk galaxies (3391 velocity measurements), with the stellar mass-to-light ratio free per galaxy and a systematic error floor scanned over 0—15%. At the reference 10% floor, the cored models give lower total χ^2 than NFW and both MOND forms: χ^2/N = 0.57 (2D), 0.57 (Burkert), 0.60 (PI), versus 0.79 (NFW), 1.38 (MOND-simple) and 1.66 (MOND-standard). The three cored models (2D, Burkert, PI) have similar goodness of fit, but the information criteria moderately favour the 2D model over PI (∆AIC ≃ 93) and provide only a modest preference over Burkert (∆AIC ≃ 13); the present data do not establish a decisive physical distinction among them. The discriminating observable is the shape of the transition to the flat asymptote: 1/R^2 for the 2D potential, 1/R for PI, and a peak followed by a slow decline for Burkert. We show that SPARC covers the relevant radial range only partially (23% of data points in the discrimination zone). Against MOND the result depends on the model-selection criterion, because MOND has substantially fewer global parameters. Among the models evaluated at each error-floor value, the 2D model gives the lowest total χ^2; MOND-standard is evaluated only at the reference 10% floor. We conclude that the soft-logarithmic potential is a viable, normalised member of the cored-model family, with a specific falsifiable prediction for the transition region.

In this paper, actions and language are closely bound together. Action sequences could be coded and act as fundamental building block of language. Action language describes the world. Thus, the world has been mapped into action space.Language Image Natural Modeling Architecture (LINMA) contributes laws to construct the mapping engineering: law of head, law of limb posture, law of hand shape, law of motion trajectory.Language token could be action token. Action language could directly drive humanoid robot to perform actions.

A derivation, once completed, is typically treated as settling the question it was undertaken to answer. This paper argues that this is not generally true, and that the gap between a completed derivation and a settled physical question is easy to overlook precisely because it has no name and leaves no formal trace once crossed. We develop the point through a case in which the gap is unusually visible: a citation, in Hawking’s canonical 1976 physics paper, offered as the source of an extension the citing paper needs but does not itself derive, which turns out oninspection to point to two unsupported sentences in an earlier paper. We use this case not toadjudicate the underlying physical claim, but to isolate two distinct achievements that citationpractice, and derivation itself, can silently conflate: extending a mathematical formalism to abroader domain, and showing that the extended formalism answers a question about a physicalsystem, in the operational sense that connects a model’s terms to what an observer would measure. We argue, drawing on Bridgman’s operational analysis and the model/target-system distinction in philosophy of science, that even a fully executed derivation closing the citation gap we identify would establish the first achievement without thereby establishing the second: a frequency-resolved extension of Hawking’s formalism cannot characterize the polarization structure of the emitted state, and so cannot by itself be used to evaluate the unitarity question the citation is invoked to support, whatever that question’s true answer turns out to be. The dependency this creates is not merely evidentiary but constitutive of Hawking’s own 1976 formal apparatus: that paper’s superscattering operator is built from Hilbert spaces the paper does not construct for the higher-spin sectors, so its own formal question, for those sectors, remains a construction the citation was invoked to supply and does not. We do not argue that this precise structure recurs throughout physics; doing so responsibly would require the same case-by-case verification this paper insists on. We do argue that the case examined here is nota bibliographic curiosity but an instance of a general and underappreciated distinction in whatclosing a derivation can and cannot be shown to achieve.

The Rindler Horizons are known to causally disconnect regions of the flat spacetime from an accelerating observer. In this article it is shown that the past and future horizons of such observer indeed meet at the event horizon of the Unit Black Hole (UBH) of the c-SRQM theory. Furthermore, the Unruh temperature registered by an accelerating observer is shown to be the red shifted Hawking temperature of the UBH at such a hovering distance where the local UBH gravity is equal to the acceleration of the observer. Thermodynamics of the gravity and acceleration are unified.

Time is conventionally treated as a fundamental parameter of physical reality.We propose an alternative axiomatic framework in which the universe is postulated as a closed, maximally entangled quantum many-body system with nofundamental notion of time. In this framework, observers are finite quantumsubsystems with limited information-processing capacity, preventing access tothe complete universal state. We argue that the perceived flow of time emergesfrom observer-dependent reconstruction of incomplete information rather thanfrom an intrinsic temporal dimension. This perspective establishes a timelessinformation-theoretic description of the universe and provides a mathematicalfoundation for investigating the emergence of temporal order, observation, andclassical reality within a unified quantum framework.

We study a self-contained combinatorial equality, discovered by the author some forty years ago, which extends [(n, p)^T)]to negative integer indices without invoking any extension of the factorial...

An addition chain of length $h$ that leads to an integer $ngeq 2$ is a strictly increasing sequence of positive integers $s_0=1,s_1=2,ldots,s_h=n$ such that $s_i=s_j+s_k$ with $i>jgeq kgeq 0$ for each $iin {1,ldots,h}$. We denote the minimal length of an addition chain that leads to an integer target $cdot$ by $ell(cdot)$. Combining a generalized version of the Brauer method with a multi-step minimization approach, we verify the speculative inequality $$ ell(2^n-1)leq n-1+ell(n) $$for all $ngeq 2$.

In verifier-style RLVR, group-relative optimization often treats advantage scale as an implementation detail. This paper separates two low-variance cases: sub-resolution jitter that should not become a preference signal, and credible but small cardinal gaps that should be learned without distorting KL calibration. We propose an advantage-scale three-way calibration interface: the same within-group scale denominator simultaneously determines the reward-branch strength, prompt-level batch weight, and the effective KL calibration induced when the reward branch is re-expressed on the original cardinal scale. This interface explains why RLOO / Dr.GRPO can let credible small gaps become KL dominated, whereas GRPO's standard-deviation denominator can amplify tiny gaps without bound. Based on this interface, we further introduce the Reward-Resolution Protocol and MaxNorm-AC, respectively filtering sub-resolution gaps and providing bounded cardinal recovery on credible nonzero gaps. Across dense / MoE architectures and math / code reasoning, MaxNorm-AC improves over the prespecified percentile-scale reference (p = 90, hereafter the p90 reference) while truncating the low-variance inverse-scale tail.

We present a formulation of Cosmological Special Theory of Relativity(CSTR) in with the standard Lorentz transformation is extended to incorporate the expansion of the universe via the Robertson-Walker scale factor.

Background: Adolescent depression has reached epidemic proportions in urbanChinese cohorts, with prevalence estimates exceeding 20% in some regions. Standard interventions achieve only partial remission in a substantial minority of cases,and treatment resistance is especially common when family-system dynamics perpetuate the depressogenic environment.Case Presentation: We report the case of "Z.," a 17-year-old male adolescent withmajor depressive disorder, generalized anxiety, school refusal, and disordered gaming, who had declined pharmacotherapy and electroconvulsive therapy. The patientpresented following academic setback, romantic dissolution, and familial conflict exacerbated by the COVID-19 pandemic.Intervention: Treatment consisted of 18 months of weekly video-based psychotherapy grounded in the Universal Law of Subconscious State Change (ULSSC), delivered concurrently to the patient and his mother. The intervention integrated cognitive training, behavioral activation, and family-systems restructuring.Results: The patient resumed educational engagement, achieved university admission, and demonstrated sustained functional recovery at 18-month follow-up. Depressive symptomatology reduced from severe to subclinical levels; suicidal ideationresolved by month four and did not recur.Conclusion: This case illustrates the potential efficacy of family-integrated subconsciousstate therapy for treatment-resistant adolescent depression. Controlled trials arewarranted.

In this paper, we study the Fifa World Cup 2026 matches up to the Final. In the Group Stage, for the decided matches, the probability for a team with lower relative Gini coefficient to win was 28/52. In the next stages up to final i.e. in the Round of Thirty Two stage, the Round of Sixteen stage etc, the probability for a team with lower relative Gini coefficient to win were 6/15, 6/8, 1/4, 0/2, 0/1, 1/1 respectively. In the Group Stage, the probability to win with lower HDI was 19/52 i.e. less than half. In the Round of Thirty Two stage, theprobability to win with lower HDI was 6/16.In the Round of Sixteen stage, the probability to win with lower HDI was 1/8. In the Quarter Final stage, the probability for a team with lower relative HDI to win was 3/4 i.e. swung to the right. In the Semi Final stage, the probability for a team with lower relative HDI to win was 2/2. In the Third Place Play-off, the team with higher relative HDI won i.e. the probability swung to the left. In the Final, the team with lower relative Gini coefficient, higher HDI won. The overall probability to win with lower relative Gini coefficient was 42/53 i.e. 0.506 i.e. more than half. In the previous World Cups 2022, 2018, 2014, the overall probability to win with lower relative Gini coefficient were 0.556, 0.627, 0.519 respectively.

The string tensions can be dynamical as one of us (EG) has studied in recent publications. For example, in the case when string theories are formulated in the modified measure formalism where string and brane tensions appear as an additionaldynamical degree of freedom. It is found that the tension values assigned to these strings and branes may not fixed (universal), but rather that each string and brane acquires its own tension with a different value for each string and brane. We havedisplayed cases where the spontaneous breaking of target space scale invariance is produced in the process of string-tension-generation, except for some limits where the tensions dynamically approach infinity. The introduction of internal gauge fieldsgoverns the dynamical tensions of the strings, and specific sources for these gauge fields provide diverse tensions. In this work it is shown that in special cases where the tensions have the exponential form exp(ασ1) or exp(ασ0), in the conformal gauge, the target space scale invariance can be restored by considering shifts in either the (spatial) σ1 or (temporal) σ0 string world-sheet coordinates. In the first case, the (open) strings must be of infinite length and cannot have boundaries in order to respect target space scaleinvariance. While in the second case we can consider closed or open strings without breaking target space scale invariance. World sheet conformal invariance implies a Ricci flat condition for an effective metric and which is consistent with the expected equations of motion of the strings. Boundary conditions can be introduced by introducing charges that couple to the internal gauge fields. Point like sources break the scale invariance point-wise which can lead to very interesting consequences, like the construction ofsemi-open strings which are strings which connect to a charge only at one point and where along the other direction the tension decays. These serve as models of deconfined quarks. Generalizations of these effects to brane like solutions in Milne and Wesson spaces, etc. are also discussed.

In 1859, Georg Friedrich Bernhard Riemann had announced the following conjecture, called the Riemann hypothesis : The nontrivial roots (zeros) $s=sigma+it$ of the zeta function, defined by: $$zeta(s) = sum_{n=1}^{+infty}frac{1}{n^s},,mbox{for}quad Re(s)>1$$ have real part} $sigma= 1/2$. In this paper, I give the proof that $sigma= 1/2$ using an equivalent statement of the Riemann hypothesis: the Dirichlet eta function and the functional relation verified by the zeta function.

This is an introduction to probability, written with a quantum idea in mind, namely that the semicircle law comes first. We first discuss discrete probability, notably with the binomial and hypergeometric laws, positive and negative, and the Poisson and compound Poisson laws. Then we get into the continuous case, with the basics of the theory explained, and with as starting examples the exponential, semicircle and beta distributions. Afterwards, we investigate the central limits and normal variables, both real and complex, and with a look at Rayleigh laws, and hyperspherical laws too. Finally, we discuss a number of more specialized distributions, and more specialized techniques too, and we end with an introduction to random matrices and freeness.

This paper proposes a fundamentally new approach to investigating the structure of the set of natural numbers based on the concept of alternative and connected numerical sets, one of which is the set of prime numbers. The method of dynamic sieves is described. By utilizing the proposed kinematic model and the theory of connected sets, a theorem is proven on the infinity of sets of n-tuple primes of arbitrary length and configuration. While the theory of connected sets expands the theorem, the theory of alternative sets deepens it, rendering the theorem global.

Companion reviews of the Cosmological Special Theory of Relativity(CSTR) have repeatedly identified the extension of the equal-time Klein-Gordon field quantization to a genuinely two-time(multi-epoch) propagation as the central open problem of the theory's quantum sector(Yi,2020,2021,2025).

We develop a residue-based method for recovering the coefficients of aDirichlet series from the singularities of its analytic continuation. Startingfrom the classical vertical-line inversion formula, we introduce a conformaltransformation that converts coefficient extraction into a contour integraland yields a spectral representation in terms of poles and residues. Weapply this framework to the von Mangoldt, prime characteristic, Möbius,and Euler totient functions, obtaining representations involving the zerosand poles of the Riemann zeta function. We also introduce generalized vonMangoldt functions associated with arbitrary Dirichlet series and extendthe approach to finite and Dedekind zeta functions, leading to formulas forpoint counts, prime-ideal counting functions, and characteristic functionsover number fields. Finally, we examine prime-pair, Goldbach, andprime-tuple counting functions and derive conditional error estimatesunder the Riemann hypothesis. These results present contour integrationand zeta-function factorizations as a unified mechanism for translatinganalytic spectral data into arithmetic information.

Supervised Neural Gas (SNG) introduces the margin-optimized cost function of Generalized Learn-ing Vector Quantization (GLVQ) equipped with neighborhood cooperation from Neural Gas (NG).This yields a highly interpretable learner that is less sensitive to the prototype initialization problem observed in the LVQ family of classification algorithms. Although Hammer, Strickert and Villmann presented the SNG framework as an extension applicable to any differentiable dissimilarity measure, the diagonal relevance weighting adaptation remains the only scheme to have been explored. In this paper, we investigate extensions of SNG with full matrix, local matrix, tangent and class-wise matrix distances by examining how rank-based cooperation interacts with metric parameterization learning. We provide gradient-level analysis of two failure modes: (I) gradient dilution and (II) subtraction amplification, which explain learning behavioral challenges in cooperation-based metric learning. We propose a Supervised Class-Wise Matrix Neural Gas variant (SCMNG) that identifies three structural conditions under which rank-based metric relevance learning can be achieved in light of these recorded failure modes. Ablation experiments against non-cooperative reference models indicate that the impact of rank-based cooperation depends on initialization quality, metric structureand class-wise conditions. When prototypes are placed in a representative manner, NG coopera-tion mainly results in gradient interference. In contrast, with poor initialization, NG cooperationfacilitates the distribution of prototypes across class regions.

This study is about applying the fundamentals of physics using the concepts of friction to understand and describe dance shoes and dance floor interaction. We have used sliding friction, skidding friction and rotational friction to describe our model independent study. Since we are performing a general and model independent study so we have taken the dancer to be of a uniform cylindrical shape. The various frictions used are time dependent.

Background Literature screening constitutes a critical component in evidence synthesis; however, it typically requires substantial time and human resources. Artificial intelligence (AI) has shown promise in this field, yet the accuracy and effectiveness of AI tools for literature screening remain uncertain. This study aims to evaluate the performance of several existing AI-powered automated tools for literature screening.Methods This diagnostic accuracy study employed a cohort to evaluate the performance of five AI tools—ChatGPT 4.0, Claude 3.5, Gemini 1.5, DeepSeek-V3, and RobotSearch—in literature screening. We selected a random sample of 1,000 publications from a well-established literature cohort, with 500 as randomized controlled trials (RCTs) group and 500 as others group. Diagnostic accuracy was measured using several metrics, including the false negative fraction (FNF), time used for screening, false positive fraction (FPF), and the redundancy number needed to screen.Results We reported the FNF for the RCTs group and the FPF for the others group. In the RCTs group, RobotSearch exhibited the lowest FNF at 6.4% (95% CI: 4.6% to 8.9%), whereas Gemini exhibited the highest at 13.0% (95% CI: 10.3% to 16.3%). In the others group, the FPF of the four large language models ranged from 2.8% (95% CI: 1.7% to 4.7%) to 3.8% (95% CI: 2.4% to 5.9%), both of which were significantly lower than RobotSearch's rate of 22.2% (95% CI: 18.8% to 26.1%). In terms of screening efficiency, the mean time used for screening per article was 1.3 s for ChatGPT, 6.0 s for Claude, 1.2 s for Gemini, and 2.6 s for DeepSeek.Conclusions The AI tools assessed in this study demonstrated commendable performance in literature screening; however, they are not yet suitable as standalone solutions. These tools can serve as effective auxiliary aids, and a hybrid approach that integrates human expertise with AI may enhance both the efficiency and accuracy of the literature screening process.

Modified Newtonian Dynamics (MOND) generally resolves the need for dark matter in galaxy rotation curves introducing a single new constant of acceleration $a_0$. It is well known that increasing $a_0$ by a factor of a few can alleviate the residual mass discrepancies that MOND leaves in galaxy clusters. Within a parameter-free Machian interpretation of MOND, in which $a_0sim GM_u/R_u^2$ arises from the scalar sum of inverse-square distance gravitational mass contributions in the universe, we promote $a_0$ to a variable influenced by mass external to a locally enclosed region in the spherically symmetric case. Instead of a boost of $a_0$ in terms of gravitational potentials as in EMOND, we show that a boost in terms of this directionless inverse-square field roughly amounts to the boost needed to accommodate the mass discrepancies of MOND in galaxy clusters. We conclude by beginning to generalize the proposed formulation beyond spherical symmetry.

An addition chain of length $h$ that leads to an integer $ngeq 2$ is a strictly increasing sequence of positive integers $s_0=1,s_1=2,ldots,s_h=n$ such that $s_i=s_j+s_k$ with $i>jgeq kgeq 0$ for each $iin {1,ldots,h}$. We denote the minimal length of an addition chain that leads to an integer target $cdot$ by $ell(cdot)$. In this paper, we introduce the concept of emph{quasi closed} addition chains and show that numbers $ngeq 2$ that admit a minimal-length quasi closed chains that are also a minimal-length addition (quasi complete numbers) satisfy the Scholz conjecture, precisely the inequality $$ ell(2^n-1)leq n-1+ell(n).$$

We analyze one coupling of single-site Glauber dynamics for proper (q)-colorings of the cycle (C_n) in two ways.Drift on the Hamming disagreement count and path coupling on Hamming-adjacent pairs give the one-step contractions[mathbb{E}bigl[d_{H}(X_1,Y_1)bigr]lealpha_{mathrm{cl}}(q,n),d_{H}(X_0,Y_0),qquadmathbb{E}bigl[d_{G}(X_1,Y_1)bigr]lealpha_{mathrm{pc}}(q,n),d_{G}(X_0,Y_0),]with (alpha_{mathrm{cl}}(q,n)=1-(q-4)/bigl(n(q-2)bigr)) and (alpha_{mathrm{pc}}(q,n)=1-(q-6)/bigl(n(q-2)bigr)), both attained. The thresholds are(qge5) and (qge7), and the constants separate by (2/bigl(n(q-2)bigr)) uniformly in (n). The gap is geometric.Transposing two colors across an edge produces a pair at Hamming distance (2) that no single recoloring connects,so the path metric (d_{G}) of the color graph charges a created disagreement (+2) rather than (+1); such pairspack (lfloor n/2floor) to a cycle, giving (operatorname{diam} d_{H}=n) and (operatorname{diam} d_{G}=lfloor3n/2floor) for(qge5), hence closed (O(nlog n)) bounds for both methods. The loss is attributable to the sparse edge setrather than to path coupling, since the complete edge set recovers (alpha_{mathrm{cl}}). Exact enumeration on (C_4)confirms every bound and gives worst-start (t_{mathrm{mix}}(0.05)=57,29,22,19,18) for (q=3,dots,7).

Physicists have been searching for a vibrational interpretation of the Schrödinger equation for a century; so far with limited success. This work presents a speculative vibrational interpretation in the special case of the Schrödinger equation for a hydrogen-like atom. Solutions of this equation are interpreted as high-frequency stationary vibrations of the atom’s electromagnetic field driven by de Broglie’s internal clock of the atom’s electron, which is assumed to orbit around the atom’s nucleus. The proposed vibrational interpretation challenges existing interpretations of the Schrödinger equation in various ways, which pose interesting research questions for future work.

This paper presents a proof of Fermat's Last Theorem using the symmetry properties of the cosine rule. It is demonstrated that the structural invariance of the cosine rule prevents any transformation that would yield nontrivial integer solutions for exponents greater than two. The argument establishes that no such solutions exist, thereby confirming Fermat's assertion.

This paper introduces an alternative, deterministic paradigm for the micro-world by modeling the hydrogen atom as an open, non-linear dissipative system embedded within the hydrodynamic substratum of Wheeler’s quantum foam. By replacing the abstract complex-valued wave function of standard quantum mechanics with a modified three-dimensional Van der Pol system formulated via Hamilton’s quaternions (H), we resolve the fundamental paradoxes of stationarity, instantaneous quantum jumps, and wave-particle duality. In this framework, the stable ground state of the atom emerges naturally as a stable limit cycle (attractor), where the classical Coulomb potential acts as an active negative-friction energy pump that balances velocity-dependent radiative dissipation at the Bohr radius (a0). We present a non-quantum, electrodynamic derivation of the Bohr radius and link the emission frequency directly to radiation intensity via the characteristic impedance of free space (Z0) without invoking Planck’s constant (h). Furthermore, the four traditional quantum numbers (n, l, m, s), the Zeeman splitting, and the Pauli exclusion principle are decoded as explicit geometric and topological properties of a phase-locked spatial rotator, bypassing the necessity of both the Schrödinger probability density and the complex Dirac matrices.

Graph theory plays a vital role in the study of mathematical structures and their applicationsin communication networks, optimization, and computer science. Among the variousdomination parameters, the Double Roman Domination Number has attracted significant attentiondue to its enhanced defensive strategy compared with classical domination concepts.This paper investigates the Double Roman Domination Number of Degree-Centric Graphsobtained from several standard graph families. The degree-centric transformation is firstconstructed for each graph, and the corresponding Double Roman Domination Numbers aredetermined. Based on these results, general observations and formulas are established fordifferent classes of graphs, illustrating the relationship between the degree-centric transformationand the domination parameter. The obtained results extend the study of dominationin transformed graphs and provide a foundation for future research in degree-centric graphtheory.

It is argued that the known physical fields themselves can be comprehended as extra dimensions attached to spacetime. This is achieved by copying the basic structure of Special Relativity related to objects, and pasting it at the level of fields. The role of the conversion factor played by the velocity of light there is played by Newtons constant here. This leads to a permutation of the roles of established concepts.

The Initial Entropy Problem asks why the early universe began in an exceptionally low gravitational entropy state despite matter and radiation being at thermal equilibrium. Standard cosmology accepts Penrose's Past Hypothesis as a brute empirical fact, without a dynamical mechanism that enforces it. This paper presents a candidate Center Origin White Hole Cyclic Cosmology, built from thirteen explicit axioms, offered as an axiomatic framework, not a resolution of the problem it addresses. The model proposes that the Big Bang is the time-reversed quantum-gravity bounce of a single, monolithic universal black hole. Three axioms carry the model's real explanatory weight and are flagged plainly rather than hidden in notation: Axiom 9 stipulates an explicit exception to the Second Law of Thermodynamics at the transition; Axiom 11 stipulates repulsive anti-gravity that cannot be derived from time-reversal of the Einstein field equations and is grounded only in current untestability; Axiom 13 stipulates that dark energy is dynamical rather than a true cosmological constant, an explicit, falsifiable bet on an open question in dark energy research that current measurements do not support under a constant &Lambda. The paper derives the model's recollapse dynamics as an exact consequence of the Friedmann equation for a dust-dominated closed universe and shows explicitly where this derivation fails to match a universe dominated by dark energy. The framework does not derive low initial entropy from more fundamental principles; it relocates the Past Hypothesis to a named, falsifiable transition.

Gravitational waves in the microhertz regime represent an unexplored window into slow dynamical processes in planetary interiors and the longu2011period tidal interactions within the Solar System. Jupiter, whose rotation period of 9.925 h is known with high precision, should generate a gravitational wave near 55.97 $mu$Hz if its mass distribution departs from perfect axial symmetry. Such a signal is far below the sensitivity of existing gravitationalu2011wave detectors and is strongly Doppleru2011shifted by the synodic Earth—Jupiter motion, making direct detection challenging.Using 20 years of barometric pressure data from DWD stations, we extract Jupiter’s gravitational wave through coherent demodulation of the phase modulations induced by Earth’s orbit and by the four Galilean moons. The recovered carrier frequency exhibits SNR $approx100$ and reveals a previously unknown longu2011term frequency drift associated with Jupiter’s deep interior. The modulation indices and phases of Io, Europa, Ganymede, and Callisto are stable across 752 independent realizations, demonstrating that Earth’s atmosphere responds coherently to microhertz gravitational forcing.The measured modulation indices exceed linearized Doppler predictions by a factor of 12.7, implying a propagation speed of $vapprox 0.08 c$ for the wave along the Jupiter—Earth path. Because this path lies entirely within the Sun’s gravitational potential, our results suggest that microhertz gravitational waves may experience dispersive propagation in strongly curved spacetime. No additional longu2011term phase modulations were detected. These findings open a new observational channel for studying gravitational phenomena in the Solar System at ultrau2011low frequencies.